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1 Proceedings o the 7th World Congress he International Federation o Automatic Control RES-O-RES SLEW MANEUVER OF HREE-AXIS ROAIONAL FLEXIBLE SPACECRAF Jae Jun Kim and Brij N. Agrawal 2 Naval Postgraduate School, Monterey, CA his paper presents a slew maneuver control design o three-axis rotational lexible spacecrat. he ocus o the work is to investigate the nonlinear eect o the three axis maneuver or a lexible spacecrat when a vibration suppression technique or linear systems such as input shaping is used in the control design. A simple method o slewing three-axis rotational spacecrat using input shaping is proposed and the proposed technique is implemented on an experimental three-axis spacecrat simulator. Keywords: High Accuracy Pointing, Guidance, Navigation and Control o Vehicles. INRODUCION For spacecrat applications where rapid slew maneuvers or the acquisition o a target point are required, the interaction o lexible structures with the spacecrat attitude control system can signiicantly degrade perormance in terms o pointing accuracy and acquisition time. With the current trend in spacecrat design o increased lexibility, multiple bodies, and stringent attitude control accuracy requirements, conventional control techniques may not be adequate to meet perormance requirements o uture spacecrat missions. Developing controls or lexible spacecrat is an especially challenging task. For point-to-point slew maneuvers or Line o Sight (LOS) spacecrat, the residual vibration at the end o the maneuver signiicantly degrades the pointing accuracy. his residual vibration also increases slewing time especially or spacecrat with large structures and low requency modes. o properly design a slew maneuver control system, pre-planned eedorward control proiles and corresponding reerence trajectories are required. Especially or ast slew maneuvers o a lexible spacecrat, careul planning o slew trajectories and eedorward control input is a crucial element o the control design. he slew trajectories and eedorward control design is based on the mathematical model o a spacecrat. hereore, accurate modelling o a spacecrat is also important. raditionally, spacecrat are modelled as rigid bodies, and any vibration resulting rom the lexible structures are considered to be disturbances. hereore, traditional eedorward control is o the smooth bang-bang type, where rapid slew is obtained rom the nature o the bang-bang control while the disturbance rom the lexible structure excitation is reduced by smoothing the edges o the bang-bang control. It is generally known that there is a trade-o between the control actuation time and the level Research Assistant Proessor, jki@nps.edu 2 Distinguished Proessor, agrawal@nps.edu o lexible structure excitations with this approach. hereore, it may require unnecessarily large control actuation time to obtain the required level o vibration suppression. Instead o reducing structural excitations with the smooth control proile, the input shaping technique attempts to cancel structural vibrations by convoluting a reerence input with an impulse sequence. With the exact knowledge o the system requencies, targeted structural requencies at the end o the maneuver can be completely eliminated in a inite time. he input shaping technique was developed or linear systems and thereore, direct application to the three axis rotational lexible spacecrat problem is somewhat limited. For spacecrat problems with slow maneuvers, the non-linear coupling between three axes is relatively weak. Much previous work has utilized either linearization or small angular rate assumptions to apply input shaping techniques. Furthermore, the linear model can be decoupled (Eigen-axis rotation) and written in modal coordinates, which permits us to examine it as a single-axis rotational spacecrat problem. In this paper, we attempt to include a nonlinear coupling term between the three axes in the design o the reerence eedorward control proiles. Input shaping technique is also redesigned or this coupled three-axis rotational spacecrat problem. he proposed technique is implemented on the experimental three-axis rotational spacecrat test-bed. 2. BACKGROUND A rigid body spacecrat can be represented in the spacecrat body coordinates as J & ω + ω Jω = Bu () where J is a inertia matrix, ω is a angular rate, u is a control input, and B is a control inluence matrix. he cross product term in Equation appears nonlinearly in the equation. For slew maneuvers with large angular rates during the maneuver, eect o the nonlinear term may become signiicant. hereore, it may be useul to include the non-linear eect when designing control proiles or /8/$2. 28 IFAC /2876--KR-.26

2 three-axis rotational spacecrat. A method to determine control proiles including the nonlinear coupling or rigid body spacecrat is previously proposed by Bell and Junkins (994). he idea is to irst design slew trajectories using linear model, then design a control proile which ollow the trajectories. he nonlinear coupling is included when the control proile is determined. For designing slew trajectories as a irst step, simple uncoupled linear model is used. he equation o motion o three-axis uncoupled rigid bodies is written as J && θ = Bu (2) where J is a diagonal inertia matrix, B is a diagonal control inluence matrix, and θ represents the state o the reerence trajectory in the global coordinates. u max -u max Figure : Smooth Bang-o-Bang Proile Example he control input can be designed to meet the slew angle required separately or each axis, which can be written as u = umax (, t t (3) where u is the maximum available control or control max design and (, t t, α ) is a unction typically represent a smooth bang-o-bang type control as shown in Figure. he o period is due to momentum saturation (or momentum transer actuators) or uel consumption constraints (or jet thrustering actuators). his smooth bango-bang is a unction o smoothing parameter ( α ) the inal time ( t ). It is noted that the inal eedorward control input may be larger than u max because the inal control input is designed to ollow the trajectories designed with assumed uncoupled linear model. hereore, there should be some margin between maximum control input or control design and the actual maximum control input with this approach. With the control input shown in Equation 3, the reerence angle and angular rate trajectories in the global coordinates or the uncoupled equation o motion become && θre () t = J Bumax (, t t t & θ () t = & (4) θ + J Bu ( τ, t dτ re max t τ & max θ () t = θ + θ t+ J Bu ( η, t dηdτ re t A t For rest-to-rest maneuvers, the total maneuver angle becomes t t τ max (,, ) θ θ J Bu η t α dηdτ he Equation can be solved or the total maneuver time required or each axis denoted by t, t, t. For actual = () 2 spacecrat with coupling between axes, the total maneuver time or all three axes should be identical; otherwise the boundary condition on slew angle and angular rate (zero or rest-to-rest maneuver) will not be satisied or axes with smaller total maneuver time. he total maneuver time or all three axes is simply t = max( t, t, t ) (6) 2 3 his total maneuver time is used to modiy the control proile to meet the slew angle and angular rate boundary conditions. A simple method is to modiy the maximum torque required on each axis, which can be computed as u B J( θ θ) R = (7) t τ ( η, t dηdτ It should satisy the slew angle and angular rate constraints at the inal time or all three axes when this modiied control input, u = ur (, t t, is applied to Equation 2. he reerence trajectories resulting rom solving Equation 2 becomes && θre () t = J BuR (, t t t & θ () t = & θ + J Bu ( τ, t dτ (8) re R t τ re = + & + R θ () t θ θ t J Bu ( η, t, α ) dηdτ Equation 8 has a common inal time or all three axes whereas Equation 4 does not. he reerence trajectories shown in Equation 8 are represented in the inertial rame. In order to convert the inertial rame reerence into the spacecrat body rame where the control input is typically represented, the ollowing relationship is used. where, ( ) re ωre () t = C( θ )& re θre (9) d & ωre ( t) = C( θre ) & θre + C( θre )&& θre dt C θ transorms the inertial angular rate ( & θ ) into the body angular rate ( ω re ) with respect to the inertial rame. he reerence control proile can be determined rom rigid-body spacecrat model in Equation. ure () t = B ( J & ωre + ωre Jωre ) () he resulting reerence control input is used as a eedoraward control to the spacecrat. his reerence control proile satisies rest-to-rest maneuver boundary constraints or rigid spacecrat with nonlinear coupling. However, when the lexibility is present in the spacecrat, residual vibrations will remain at the inal time. 4. INPU SHAPING DESIGN FOR HREE-AXIS ROAIONAL FLEXIBLE SPACECRF 3 re 2

3 It is previous shown in many literatures including Kim and Agrawal (26) that input shaping techniques can improve the perormance o slew maneuvers or single-axis lexible spacecrat simulator. With the input shaping technique, the reerence control input is convoluted with a sequence o impulses to cancel vibrations rom lexible modes. his is equivalent to cancelling the lexible poles by adding a timedelay which has zeros at the lexible pole locations. For a three dimensional rotational spacecrat, the equation o motion becomes nonlinear and it is unable to determine the requency o the system (i.e. eigenvalues). hereore, the system must be linearized to apply the input shaping technique. he equation o motion o the system can be linearized with respect to the inal state o the system proposed by. Singh, and S.R. Vadali (993), which has shown to be eective. Equation o motion o a lexible spacecrat can be written as J & ω+ ω Jω+ D && η = Bu () && η+γ & η+λ η+ D & ω = where J is the inertia matrix, ω is the angular rate o the spacecrat body, D is a matrix representing rigid-elastic coupling, η is the lexible state vector in generalized coordinates, Γ is the diagonal damping matrix, and Λ is a diagonal matrix representing the requencies o the lexible body. Equation is written in hybrid coordinates. he linearized equation o motion is simply written as J & ω+ D && η = Bu (2) && η+γ & η+λ η+ D & ω = he system requencies or input shaping technique are determined rom Equation 2. he irst step or reerence control proile design is to determine the reerence trajectories. Starting with the uncoupled rigid body equation, J && θ = Bu, the reerence trajectories can be similarly determined as shown as Equation 8 with a control input o u = ur (, t t. One option is to compute the reerence input as shown in Equation irst, and then shape the resulting reerence input. However, i the input shaper is directly used to shape the reerence input in Equation, considerable rigid body position and velocity error will remain at the inal time, which leads to the system states driting away rom the desired inal state. Instead o shaping the reerence input directly, the proposed development will shape the control input o u = ur (, t t. his ensures that the resulting reerence trajectories will satisy the boundary constraints o a slew maneuver. he shaped torque proile becomes n ushaped = Aiδ ( t ti )* ur ( t, t, α ) (3) n i= where Aiδ ( t ti) represents the impulse sequence or i= input shaping, and * represents the convolution operation. When the reerence trajectory is created with the shaped control input, uncoupled rigid body equation, J && θ = Bu, can be used since it will still satisy the constraints. However, it is more advantageous to use the linearized lexible spacecrat equation shown in Equation 2 to create a trajectory. he resulting reerence trajectories rom Equation 2 and Equation 3 should still satisy the boundary constraints o a slew maneuver while more closely represent the actual spacecrat system than simple uncoupled system. In addition, the reerence trajectories results rom Equation 2 and Equation 3 will include lexible mode state (η,η& ) trajectories. Using the same kinematic relationship as mentioned in the previous section, ωre () t = C( θ )& re θre (4) d & ωre ( t) = C( θre ) & θre + C( θre )&& θre dt the reerence control proile including input shapers can be computed as ure () t = B ( J & ωre + ωre Jωre + D && η) () In the development o reerence trajectories and control proile, linearization o the system model and recomputation o reerence control rom shaped control input will hinder rom perect cancellation o lexible modes. he robust version o the input shapers such as zero-vibrationderivative (ZVD) shapers are highly desirable in the design.. INPU SHAPING SIMULAION he mathematical model o the lexible spacecrat is shown in Equation. he parameters used or simulation are 326 J 44, = D =, , 2(.2) ζ Λ= 2 Γ=, and.88 2(.88) ζ ζ =.. he slew requirements are 3 degrees or all three axes rom a zero attitude. he irst step is to design rest-torest control proiles or the uncoupled rigid body problem. In order to exempliy the eect o lexibility, bang-bang control proile without smoothing is used in the simulation ime (sec) Figure 2: Bang-Bang Control or uncoupled axis rotation, u = u (, t t, α =, t = R 26

4 Figure 2 shows the bang-bang control proile or each axis when the inal time is adjusted to have the same total maneuver time. When the rigid body spacecrat model is used, the reerence control input is computed rom Equation. Figure 3 shows the resulting reerence control input when the rigid body spacecrat model is considered in the design. he control input is modiied rom Figure 2 to satisy the 3 degree slew maneuvers or all three axes using Equation. When this reerence control designed or rigid spacecrat is applied to the lexible spacecrat, considerable residual vibration will result as shown in Figure 4. he maximum amplitude o the residual vibration at the end o the maneuver is close to degree rom yaw angle trajectory in Figure ime (sec) Figure 3: Reerence Feedorward Control Proile Based on Rigid Body Spacecrat Model roll(deg) ime (sec) Figure 4: Resulting rajectories with Reerence Feedorward Control in Figure 3 Figure shows the shaped input rom Figure 2 using the requencies o the system. he requencies o the system will be dierent rom the requencies in Λ matrix since the system requency is computed rom the whole system represented in Equation. he total maneuver time has increased as a result since each requency in the system will add hal o the damped period to the maneuver time. Figure 6 shows the reerence control proile when the shaped bangbang control in Figure is modiied to satisy boundary conditions using Equation. he resulting trajectories showed good results with very little residual vibration (maximum residual vibration amplitude o.3 degree) and very little rigid body angular and angular rate errors ime (sec) Figure : Input Shaped (ZVD) Bang-Bang Control Proile rom Figure ime (sec) Figure 6: Reerence Feedorward Control Proile with Input Shaping Based on Flexible Spacecrat Model roll(deg) ime (sec) Figure 7: Resulting rajectories with Reerence Feedorward Control in Figure 6 27

5 6. EXPERIMEN WIH HE SPACECRAF SIMULAOR Figure 8: NPS hree-axis Spacecrat Simulator 2 Figure 8 shows the current three-axis spacecrat simulator, which is an experimental test-bed or the Biocal Relay Mirror Spacecrat. he key components o the spacecrat simulator include Spherical Air-Bearing, PC4 Onboard Computer with CP/IP Ethernet Port, Serial Port, Analog I/Os, and Digital I/Os, Matlab/Simulink (via xpc arget) Real-ime Control Interace, IMU with integrated 3-Axis Fiber Optic Laser Gyros, Reconigurable Control Moment Gyroscope (CMG) Array, a IR sensor, a Magnetometer and 2 inclinometers, an Automatic Mass Balancing System, and Passive Saety bumpers. hree lexible mass simulators are also installed on the test-bed, which provide low requency vibration (~.Hz) or all three axes and comprise about 9% o the principal moment o inertia. A detailed drawing o each lexible mass is shown in Figure. An adjustable length torsional rod provides stiness or the lexible masses. he lexible masses can be securely locked to simulate rigid body spacecrat when desired. Beore perorming slew maneuver experiments, system identiication o the spacecrat should be perormed. When the lexible mass simulators are locked in places, the rigid spacecrat has an inertia matrix o J = (6) he value o the inertia matrix is determined using least square method using momentum balance equation. It has been assumed that the rigid-elastic coupling is zero simply because it is currently not available. All three lexible masses have requencies very close to.hz with damping values o around %. Applying these values into Equation provides the mathematical model o the current spacecrat simulator. Because we ignored the rigid-elastic coupling, it is expected that additional error will be present or slew maneuver control. Initial attitude is [ ] degrees and the inal attitude is [ 7] degrees or slew maneuver experiments. First step or slew control design is to determine control proile and trajectories or a simple uncoupled system. In the current spacecrat test-bed, three Control Moment Gyroscopes (CMGs) are used as primary actuators. his CMG array will exhibit a singularity state or certain geometric conigurations o gimbal angles. For an agile maneuver, it is highly desirable to utilize the ull momentum envelop o the CMG array without encountering singularity problems. he CMG steering law is written as h& = A( δδ ) & & δ= A( δ) h& (6) where h & is the torque rom the CMG array, & δ is the gimbal rate vector, and A( δ ) is the Jacobian matrix determining the gimbal steering law. he singularity condition corresponds to the case where the matrix A( δ ) becomes singular. When the matrix A( δ ) is singular, the gimbal rate cannot be determined rom the amount o torque required. In addition, the maximum control torque rom the CMG will be limited by the gimbal rate such that h& max = A( δ ) & δ (7) max Since the matrix A( δ ) is time-varying, the maximum available torque is not constant and Equation 3 cannot be used. Instead o using Equation 3 to determine the control torque or uncoupled system, simple parameter optimization routine is ormulated to determine the gimbal rate which satisies the slew boundary conditions. Gimbal rates o three CMGs are parameterized as versine proiles. he uncoupled system used or optimization ignores the lexibility and the cross coupling. J & dω = A( δδ ) & ω h( δ) (8) It is noted that the non-diagonal elements o the inertia matrix is also ignored to ormulate a simpler problem. he resulting gimbal rate proile which satisies the constraints are shown in Figure 9. he maximum gimbal rate is limited by. rad/s. he gimbal rate cannot be parameterized as bang-bang in real systems since CMG system cannot change the rate instantaneously. hereore, versine proile is used instead. δ dot δ dot 3 δ dot ime (sec) Figure 9: Gimbal Rate Proile or Uncoupled System he corresponding control torque proile or Figure 9 is shown in Figure. his control proile is used to create 28

6 reerence trajectories o the system. When only the rigid body is considered, the reerence control torque proile is computed as shown in Figure. In order to apply this control to our spacecrat simulator with CMGs, the required torque should be converted into a required gimbal rate command. Figure 2 shows the required gimbal rate when steering law in Equation 6 is used. At around 4 seconds in Figure 2, gimbal rate command showed a very large value which indicates that the system is singular. u Rx u Ry u Rz ime (sec) Figure : Control orque or Uncoupled System u rex u rey proiles, no singularity has encountered. his is due to the act that the amplitude o the control has decreased as a result o input shaping and the gimbal system stayed within the singularity-ree momentum space. he resulting reerence gimbal rate proile with input shaping is shown in Figure. δ dot 2 δ dot δ dot ime (sec) Figure 2: Gimbal Rate Proile or Generating Control orque in Figure u rez ime (sec) Figure : Reerence Control orque Based on Rigid Body Spacecrat Model hereore, this gimbal rate command cannot be used in its current orm. In act, when the optimization problem is solved to determine the gimbal rate proile in Figure 9, the gimbal trajectories also undergoes singular states. However, no inversion is necessary to solve the optimization problem, and smooth control proile can be obtained. On the other hand, solving or gimbal rate rom the control torque cannot be simply determined. For experiments, gimbal rate or uncoupled system will be used or residual vibration demonstration purpose. Figure 3 shows input shaped control proiles using ZVD at. Hz rom the control torque in Figure. he slew trajectories were generated and the resulting control torque proile is shown in Figure 4. Luckily, when the reerence control torque proile shown in Figure 4 is used to determine the reerence gimbal rate ime (sec) Figure 3: Input Shaped Control Proile using Control orque Proile shown in Figure ime (sec) Figure 4: Reerence Control orque Proile with Input Shaping 29

7 δ dot δ dot 2 δ dot ime (sec) Figure : Gimbal Rate Proile with Input Shaping For comparison purposes, two dierent control proiles shown in Figure 9 and Figure are applied to the spacecrat simulator. Only eedorward control is applied because eedback control may create singularity problems near singular states. In order to close a eedback loop about the reerence trajectories, the control proile could be redesigned to stay within the singularity-ree region. However, with small usage o momentum space, maneuver will be slow and lexibility and non-linear eect will not be exempliied roll (deg) Figure 6: Spacecrat Simulator Attitude rajectories with Gimbal Rate shown in Figure 9 Figure 6 Shows the attitude trajectory o the spacecrat simulator when the gimbal rate shown in Figure 9 is applied to the experimental test-bed. It showed a residual vibration (maximum amplitude o around deg) at the end o the maneuver. Figure 7 shows the results when the gimbal rate shown in Figure is applied. Even though the rigid-elastic coupling is ignored in the spacecrat model, the residual vibration is almost completely eliminated. Figure 6 showed a large steady-state angular error in yaw angle since the trajectories are based on the rigid spacecrat. Figure 7 shows some residual rigid body angular rate in yaw angle due to zero elastic-rigid coupling assumption. roll (deg) Figure 7: Spacecrat Simulator Attitude rajectories with Gimbal Rate shown in Figure 6. CONCLUSION In this paper, input shaping technique is applied to the lexible nonlinear spacecrat problems. First control proile is designed by ignoring nonlinear and lexible terms in the equation o motion. his control input is used or input shaping. he resulting shaped input is urther modiied to include the nonlinear and coupling eect. It has shown in the simulation and experiments that this method can be eective in designing slew trajectories or lexible spacecrat. Further study is necessary to compare perormances with other leading alternative techniques or slew maneuvers and complete demonstration o the techniques using the spacecrat simulator. REFERENCES B. Wie, D. Bailey. and C. Heiberg (2), Singularity Robust Steering Logic or Redundant Single-Gimbal Control moment Gyros, Journal o Guidance Control and Dynamics, Vol. 24, No.. September-October.. Singh, and S.R. Vadali (993), Input Shaped Control o hree-dimensional Maneuvers o Flexible Spacecrat, Journal o Guidance Control, and Dynamics, Vol. 6, No. 6. M. J. Bell and J.L. Junkins (994), Near Minimum-ime hree Dimensional Maneuvers o Rigid and Flexible Spacecrat, Journal o the Astronautical Sciences, Vol. 42., No. 4. G. Marguiles and Aubrun (978), Geometric heory o Single Gimbal Control Moment Gyro Systems, Journal o the Astronautical Sciences, Vol. XXVI, No. 2.. Singh and W. Singhose (22), utorial on Input Shaping/ime Delay Control o Maneuvering Flexible Structures, 22 American Control Conerence, May 8-, Anchorage, Alaska. J. J. Kim and B. N. Agrawal (26), Experiments on Jerk Limited Slew Maneuvers o a Flexible Spacecrat, AIAA Guidance, Navigation, Control Conerence, Keystone, CO. 26